Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of $BF$ theory

Abstract: We give an interpretation of the holographic correspondence between two-dimensional $BF$ theory on the punctured disk with gauge group ${\rm PSL}(2,\mathbb R)$ and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian $S^1$-action on the orbits. The partition function is given by a sum over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.